Quantum Matter

Many-body entanglement, the special quantum correlation that exists among a large number of quantum particles, underlies interesting topics in both condensed matter and quantum information theory. On the one hand, many-body entanglement is essential for the existence of topological order in condensed matter systems and understanding many-body entanglement provides a promising approach to understand in general what topological orders exist. On the other hand, many-body entanglement is responsible for the power of quantum computation and finding it in experimentally stable systems is the key to building large scale quantum computers. In this talk, I am going to discuss how our understanding of possible many-body entanglement patterns in real physical systems contributes to the development on both topics. In particular, I am going to show that based on simple many-body entanglement patterns, we are able to (1) completely classify topological orders in one-dimensional gapped systems, (2) systematically construct new topological phases in two and higher dimensional systems, and also (3) find an experimentally more stable scheme for measurement-based quantum computation. The perspective from many-body entanglement not only leads to new results in both condensed matter and quantum information theory, but also establishes tight connection between the two fields and gives rise to exciting new ideas.

I will describe our recent work on a new topological phase of matter: topological Weyl semimetal. This phase arises in three-dimensional (3D) materials, which are close to a critical point between an ordinary and a topological insulator. Breaking time-reversal symmetry in such materials, for example by doping with sufficient amount of magnetic impurities, leads to the formation of a Weyl semimetal phase, with two (or more) 3D Dirac nodes, separated in momentum space. Such a topological Weyl semimetal possesses chiral edge states and a finite Hall conductivity, proportional to the momentum-space separation of the Dirac nodes, in the absence of any external magnetic field. Weyl semimetal demonstrates a qualitatively different type of topological protection: the protection is provided not by the bulk band gap, as in topological insulators, but by the separation of gapless 3D Dirac nodes in momentum space. I will describe a simple way to engineer such materials using superlattice heterostructures, made of thin films of topological insulators. References: arXiv:1110.1089; Phys. Rev. Lett. 107, 127205 (2011); Phys. Rev. B 83, 245428 (2011)."

Topology has many different manifestations in condensed matter physics. Real space examples include topological defects such as vortices, while momentum space ones include topological band structures and singularities in the electronic dispersion. In this talk, I will focus on two examples. The first is that of a vortex in a topological insulator that is doped into the superconducting state. This system, we find, has Majorana zero modes and thus, is a particularly simple way of obtaining these states. We derive general existence criteria for vortex Majorana modes and find that existing systems like Cu-doped Bi2Se3 fulfill them. In the process, we discover a rare example of a topological phase transition within a topological defect (the vortex) at the point when the criteria are violated. The second example is that of Weyl semimetals, which are three-dimensional analogs of graphene. Interestingly, the Dirac nodes here are topological objects in momentum space and are associated with peculiar Fermi-arc surface states. We discuss charge transport in these materials in the presence of interactions or disorder, and find encouraging agreement with existing experimental data.

Dualities in physics are well known for their conceptual depth and quantitative predictive power in contexts where perturbation theory is unreliable. They are also remarkable for the staggering arrange of physical problems that exploit them, ranging from the study of confinement and unconventional phases in statistical mechanics and field theory to the unification of the string theory landscape. In this talk I will present a new, completely general approach to dualities that affords a systematic theory of quantum dualities and incorporates classical dualities as well into one unified framework. This new algebraic approach is remarkably successful in extending powerful duality techniques to the context of topologically quantum ordered systems and quantum information processing, and affords a compelling foundation for a general theory of exact dimensional reduction or holographic correspondences. Many systems however display only approximate dimensional reduction, and so I will present general inequalities -some of them based on entanglement- linking quantum systems of different spatial dimensionality. These inequalities provide bounds on the expectation values of observables and correlators that can enforce an effective dimensional reduction. In closing I will discuss some implications for (topological) quantum memories.

Weak topological insulators have an even number of Dirac cones in their surface spectrum and are thought to be unstable to disorder, which leads to an insulating surface. Here we argue that the presence of disorder alone will not localize the surface states, rather, the presence of a time-reversal symmetric mass term is required for localization. Through numerical simulations, we show that in the absence of the mass term the surface always flow to a stable metallic phase and the conductivity obeys a one-parameter scaling relation, just as in the case of a strong topological insulator surface. With the inclusion of the mass, the transport properties of the surface of a weak topological insulator follow a two-parameter scaling form.

We report on our recent progress to investigate materials classes exhibiting d+id superconductivity, where topologically nontrivial pairing phases can emerge. Specifically, motivated by recent experimental progress, we show that graphene doped to the van Hove regime can give rise to a plethora of interesting ordering instabilities such as spin density wave and superconductivity. As a function of system parameters such as doping and range of Coulomb interaction, we explain which instability is favored by the system, and analyze the effect of long-range interactions on superconductivity giving rise to a competition between singlet d+id and triplet f wave. We also outline our work in progress for other materials classes which we believe are promising to stabilize such interesting topological superconducting states of matter.

It has been well-known that topological phenomena with fractional excitations, i.e., the fractional quantum Hall effect (FQHE) will emerge when electrons move in Landau levels. In this talk, I will show FQHE can emergy even in the absence of Landau levels in interacting fermion models and boson models. The non-interacting part of our Hamiltonian contains topologically nontrivial flat band.

Recently, we developed a user friendly scheme based on the quantum kinetic equation for studying thermal transport phenomena in the presence of interactions and disorder . This scheme is suitable for both a systematic perturbative calculation as well as a general analysis. We believe that this method presents an adequate alternative to the Kubo formula, which for thermal transport is rather cumbersome. We have applied this approach in the study of the Nernst signal in superconducting films above the critical temperature. We showed that the strong Nernst signal observed in amorphous superconducting films, far above Tc, is caused by the fluctuations of the superconducting order parameter. We demonstrated a striking agreement between our theoretical calculations and the experimental data at various temperatures and magnetic fields. My talk will include a general description of the quantum kinetic approach, but mainly I will concentrate on the Nernst effect in superconducting films. I will use this example to discuss some subtle issues in the theoretical study of thermal phenomena that we encountered while calculating the Nernst coefficient. In particular, I will explain how the Nernst theorem (the third law of thermodynamics) imposes a strict constraint on the magnitude of the Nernst effect.

We investigate the entanglement spectra of topological insulators which have gapless edge states on their spatial boundaries. In the physical energy spectrum, a subset of the edge states that intersect the Fermi level translates to discontinuities in the trace of the single-particle entanglement spectrum, which we call a `trace index'. We find that any free-fermion topological insulator that exhibits spectral flow has a non-vanishing trace index, which provides us with a new description of topological invariants. In addition, we identify the signatures of spectral flow in the single-particle and many-body entanglement spectrum; in the process we present new methods to extract topological invariants and establish a connection between entanglement and quantum Hall physics in translationally invariant and disordered systems.

Time-reversal invariant band insulators can be separated into two categories: `ordinary' insulators and `topological' insulators. Topological band insulators have low-energy edge modes that cannot be gapped without violating time-reversal symmetry, while ordinary insulators do not. A natural question is whether more exotic time-reversal invariant insulators (insulators not connected adiabatically to band insulators) can also exhibit time-reversal protected edge modes. In 2 dimensions, one example of this is the fractional spin Hall insulator (essentially a spin-up and spin-down copy of a fractional quantum Hall insulator, with opposite effective magnetic fields for each spin). I will discuss another family of strongly interacting insulators, which exist in both 2 and 3 dimensions, that can have time-reversal protected edge modes. This gives a new set of examples of `fractional' topological insulators.

Strongly correlated electron systems are play grounds for exotic quantum states such as high Tc superconductivity, quantum spin liquids, non-fermi liquids and so on. Recently high Tc superconductivity has been observed in an iron based compound K2Fe4Se5. I will present a model and outline an effective theory [1] that describes physics of this complex system - a non linear O(3) sigma model in 2 + 1 dimensions coupled to Dirac fermions. Topological solitons and induced quantum numbers are well known in Skyrme model for protons and neutrons. In our context, we get topological baby Skyrmions with an induced charge 2e. Baby Skyrmions conduct themselves `super'.